Recent zbMATH articles in MSC 51E14https://www.zbmath.org/atom/cc/51E142021-04-16T16:22:00+00:00WerkzeugResults on partial geometries with an abelian Singer group of rigid type.https://www.zbmath.org/1456.510042021-04-16T16:22:00+00:00"De Winter, Stefaan"https://www.zbmath.org/authors/?q=ai:de-winter.stefaan"Kamischke, Ellen"https://www.zbmath.org/authors/?q=ai:kamischke.ellen"Neubert, Eric"https://www.zbmath.org/authors/?q=ai:neubert.eric"Wang, Zeying"https://www.zbmath.org/authors/?q=ai:wang.zeyingA partial geometry \(pg(s,t,\alpha)\) is a finite partial linear space such that every line contains \(s+1\) points, every point lies on \(t+1\) lines, and given a point \(P\) not incident with a line \(L\), there are exactly alpha lines through \(P\) meeting \(L\).
The partial geometry is said to admit a Singer group if its automorphism group contains a subgroup acting sharply transitively on the points. If the Singer group is an abelian group G, then it was shown in [\textit{S. De Winter}, J. Algebr. Comb. 24, No. 3, 285--297 (2006; Zbl 1106.51003)] that the stabiliser of a line in G is either trivial or has size \(s+1\). The authors call the partial geometry, together with the group G, of ``rigid type'' if all lines have a trivial stabiliser.
The only known example of a partial geometry of rigid type is due to Van Lint and Schrijver and has parameters \((5,5,2)\). In this paper, the authors study the possible parameters of partial geometries of rigid type (and their associated Singer group). By use of different necessary conditions on the parameters and some properties of the associated group, and aided by a computer, the authors show that if \(\alpha\leq 1000\), there are only five parameter sets that could possibly correspond to a partial geometry of rigid type.
The hypothetical partial geometries would have one of the four sets of parameters
\((11,23,3), (39,39,15), (272,272,104)\) or \((2295,4591,615)\).
The authors resist the temptation to conjecture that the Van Lint-Schrijver partial geometry is the only partial geometry of rigid type. Instead they formulate the weaker conjecture that for a partial geometry of rigid type, the number of points is either a power of \(2\) or a power of \(3\). Note that this conjecture does not rule out the existence of a partial geometry of rigid type with parameters \((11,23,3)\) or \((39,39,15)\).
Reviewer: Geertrui Van de Voorde (Canterbury)